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3D Rotation and Translation for Hyperbolic Knowledge Graph Embedding

Yihua Zhu, Hidetoshi Shimodaira

TL;DR

This work tackles the challenge of learning KG embeddings that simultaneously capture a broad spectrum of relation patterns, including symmetry, antisymmetry, inversion, both commutative and non-commutative composition, hierarchy, and multiplicity. It introduces 3H-TH, a framework that combines 3D rotation in hyperbolic space with translation in hyperbolic space, using per-relation curvature to encode hierarchy and hyperbolic geometry to preserve hierarchical structure in a low-dimensional setting. Through extensive experiments on WN18RR, FB15K-237, and FB15K, 3H-TH achieves state-of-the-art or near-state-of-the-art performance in low dimensions, while maintaining competitive results in high dimensions; analysis reveals superior handling of hierarchical and multiplicity-related patterns. The approach leverages hyperbolic embeddings, exponential and logarithmic mappings, and Möbius addition to enable rich, non-commutative relational composition, offering practical gains for link prediction in hierarchically structured knowledge graphs while acknowledging higher computational demands in hyperbolic operations.

Abstract

The main objective of Knowledge Graph (KG) embeddings is to learn low-dimensional representations of entities and relations, enabling the prediction of missing facts. A significant challenge in achieving better KG embeddings lies in capturing relation patterns, including symmetry, antisymmetry, inversion, commutative composition, non-commutative composition, hierarchy, and multiplicity. This study introduces a novel model called 3H-TH (3D Rotation and Translation in Hyperbolic space) that captures these relation patterns simultaneously. In contrast, previous attempts have not achieved satisfactory performance across all the mentioned properties at the same time. The experimental results demonstrate that the new model outperforms existing state-of-the-art models in terms of accuracy, hierarchy property, and other relation patterns in low-dimensional space, meanwhile performing similarly in high-dimensional space.

3D Rotation and Translation for Hyperbolic Knowledge Graph Embedding

TL;DR

This work tackles the challenge of learning KG embeddings that simultaneously capture a broad spectrum of relation patterns, including symmetry, antisymmetry, inversion, both commutative and non-commutative composition, hierarchy, and multiplicity. It introduces 3H-TH, a framework that combines 3D rotation in hyperbolic space with translation in hyperbolic space, using per-relation curvature to encode hierarchy and hyperbolic geometry to preserve hierarchical structure in a low-dimensional setting. Through extensive experiments on WN18RR, FB15K-237, and FB15K, 3H-TH achieves state-of-the-art or near-state-of-the-art performance in low dimensions, while maintaining competitive results in high dimensions; analysis reveals superior handling of hierarchical and multiplicity-related patterns. The approach leverages hyperbolic embeddings, exponential and logarithmic mappings, and Möbius addition to enable rich, non-commutative relational composition, offering practical gains for link prediction in hierarchically structured knowledge graphs while acknowledging higher computational demands in hyperbolic operations.

Abstract

The main objective of Knowledge Graph (KG) embeddings is to learn low-dimensional representations of entities and relations, enabling the prediction of missing facts. A significant challenge in achieving better KG embeddings lies in capturing relation patterns, including symmetry, antisymmetry, inversion, commutative composition, non-commutative composition, hierarchy, and multiplicity. This study introduces a novel model called 3H-TH (3D Rotation and Translation in Hyperbolic space) that captures these relation patterns simultaneously. In contrast, previous attempts have not achieved satisfactory performance across all the mentioned properties at the same time. The experimental results demonstrate that the new model outperforms existing state-of-the-art models in terms of accuracy, hierarchy property, and other relation patterns in low-dimensional space, meanwhile performing similarly in high-dimensional space.
Paper Structure (50 sections, 21 equations, 3 figures, 19 tables)

This paper contains 50 sections, 21 equations, 3 figures, 19 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Translation-based approach.
  4. Rotation-based approach.
  5. Hyperbolic KGE.
  6. Problem Formulation and Background
  7. Knowledge graph embedding
  8. TransE
  9. 2D and 3D rotation
  10. Hyperbolic geometry
  11. Our Approach
  12. Component models
  13. 3H-TH model
  14. Scoring function and loss
  15. Other composite models
  16. ...and 35 more sections

Figures (3)

  • Figure 1: Toy examples for three difficult relation patterns. Our approach can perform well in Hierarchy, Multiplicity, and Non-Commutative Composition.
  • Figure 2: The logarithmic transformation $\mathrm{log}_\mathbf{0}^{\xi_{r}}(\mathbf{v})$ ($\mathbb{B}^{k}_{\xi_{r}}\rightarrow\mathcal{T}^{\xi_{r}}_\mathbf{0}\mathbb{B}^{k}_{\xi_{r}}$) and the exponential transformation $\mathrm{exp}_\mathbf{0}^{\xi_{r}}(\mathbf{v})$ ($\mathcal{T}^{\xi_{r}}_\mathbf{0}\mathbb{B}^{k}_{\xi_{r}} \rightarrow \mathbb{B}^{k}_{\xi_{r}}$)
  • Figure 3: Toy examples for several relation patterns. Our approach can perform well on all these relation patterns.